Abstract and Applied Analysis

Volume 2013 (2013), Article ID 302642, 13 pages

http://dx.doi.org/10.1155/2013/302642

Research Article

Some Integrals Involving -Laguerre Polynomials and Applications

Jian Cao

Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

Received 13 January 2013; Accepted 4 June 2013

Academic Editor: Mustafa Bayram

Copyright © 2013 Jian Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The integrals involving multivariate q-Laguerre polynomials and then auxiliary ones are studied. In addition, the representations of q-Hermite polynomials by q-Laguerre polynomials and their related integrals are given. At last, some generalized integrals associated with generalized q-Hermite polynomials are deduced.

Dedicated to Srinivasa Ramanujan on the occasion of his 125th birth anniversary

1. Introduction

The -Laguerre polynomials are important -orthogonal polynomials whose applications and generalizations arise in many applications such as quantum group (oscillator algebra, etc.), -harmonic oscillator, and coding theory. For example, covariant oscillator algebra can be expressed by -Laguerre polynomials [1]. The -deformed radial Schrödinger is analyzed by -Laguerre polynomials [2]. The -Laguerre polynomials are the eigenvectors of an -representation by [3]. For more information, please refer to [1–5].

The -Laguerre polynomials are defined by [6, equation ]

which belong to the Askey scheme of basic hypergeometric orthogonal polynomials and according to Koekoek and Swarttouw [7, equation ]. The case of in (1) replaced by is studied by Moak [8, equation ].

In this paper, we first define the auxiliary -Laguerre polynomials as follows:

It is easy to see the validity of the following:

where the classical Laguerre polynomials are defined by [9, page 201]

For more information about classical Laguerre polynomials, please refer to [9–15] and the references therein.

The well-known orthogonality of -Laguerre polynomials reads the following.

Proposition 1 (see [6, equation ] and [8, equation ]). For and for , one has

Hahn [16] discovered the previous -extensions of the Laguerre polynomials, although he said little about them. Moak [8] found that the -Laguerre polynomials are orthogonal with respect to the discrete measures (Dirac measure). Koekoek and Meijer [17–19] studied systematically the inner product of -Laguerre polynomials. Ismail and Rahman [20] studied the indeterminate Hamburger moment problems related to -Laguerre polynomials. For more information, please refer to [6–8, 16–21] and the references therein.

In this paper, we first generalize Proposition 1 and the auxiliary ones as follows.

Theorem 2. For and , one has

Theorem 3. For and , one has

Corollary 4 (see [15, equation (14)]). For , , and , one has

Remark 5. Theorems 2 and 3 reduce to Proposition 1 and formula (41), respectively, if letting and , and become Corollary 4 by setting and taking .

The discrete -Hermite polynomials and are defined by [7, pages 90-91]

which are equivalent to Al-Salam-Carlitz polynomials with (please refer to [22, page 53] also), and the relation between them is . For more information about the Al-Salam-Carlitz polynomials and the discrete -Hermite polynomials, please refer to [7, 22–30] and the references therein.

In this paper, we also define new -Hermite polynomials and , whose names come from the facts

then we deduce the representations of and by -Laguerre polynomials; see Theorems 15 and 16.

As an application, using the orthogonality of -Laguerre polynomials (6), and (41), and combining the expressions of -Hermite polynomials (52) and (54), we can obtain the following results immediately.

Theorem 6. For and , one has

where is defined by (52).

Theorem 7. For and , one has

where is defined by (54).

The generalized Hermite polynomials were introduced by Szegö [31], see also [23, equation ]) as follows:

The authors [23, equation ] defined the following generalized -Hermite polynomials:

and deduced their orthogonal relations; see Proposition 19 below.

In this paper, we continue to define the auxiliary polynomials according to (16) as follows:

With the aid of (15)–(17) and (4), one readily verifies that

As another application of this paper, we gain the general -Laguerre polynomials of several variables by Theorems 2 and 3, and we also deduce the orthogonal polynomials of . For more details of the results, see Theorems 20 and 21 and Corollary 23.

The structure of this paper is organized as follows. In Section 2, we show how to prove the integrals involving -Laguerre polynomials of several variables. In Section 3, we represent discrete -Hermite polynomials by -Laguerre polynomials and their related integral results. In Section 4, we study the general integrals of -Hermite polynomials involving several variables.

2. Notations and Proof of Theorems 2 and 3

Throughout this paper, we follow the notations and terminology in [32] and assume that , , and is rational number. The -series and its compact factorials are defined [32, page 6], respectively, by

and , where is a positive integer and is a nonnegative integer or .

The basic hypergeometric series is given by

For convergence of the infinite series in (20), and when , or and when , provided that no zeros appear in the denominator. Letting and setting , (20) reduces to the classical Gauss’ hypergeometric series

where Pochhammer symbol is defined by .

The -analogue of the gamma function is defined by (see [32, equation ]) as follows:

The -Chu-Vandermonde formula [32, equations (II.6) and (II.7)] reads that

The transformations [32, equations (III.12) and (III.13)] stats that

The -analogue of the Pfaff-Kummer transformation [32, equation ] is as follows:

The Ramanujan beta integral is stated as follows [33, equation ]:

Lemma 8 (see [33, equation ]). One has

Proof. Taking in (26), then letting , we obtain (28) immediately. The proof is complete.

Lemma 9. For and , one has

Proof. Letting in [6, Proposition 4.1],

then replacing by , we have

Comparing the coefficients of on both sides of (32) yields (29). Similar to (32), by the definition (1), we have

By taking and letting in (33), we obtain (30). The proof of Lemma 9 is complete.

Lemma 10. For and , one has

Proof. Interchanging the integral and summation by definition, the left hand side of (34) equals

which is the right hand side of (34) by using the second formula of (23) and simplification. Similar to (34), the right hand side of (35) is equal to

which is equivalent to the right hand side of (35) by using the first formula of (23) and simplification. The proof of Lemma 10 is complete.

Lemma 11. If is a polynomial of degree about and is defined in the infinite interval , which can be expanded in a series of the form

where and are the th Fourier-Laguerre coefficients, and both of them are independent of , then one has

Proof. Multiplying (38) by and integrating term by term over the interval , using (6), we obtain the proof of (39). Similarly, taking in (35), we deduce

so we also gain the proof of (40). The proof of Lemma 11 is complete.

Lemma 12. For , one has

Remark 13. Replacing by and letting , we have [15, equation ]

Setting , (42) and (43) reduce to (29) and (30), respectively.

Proof. Let

By Lemmas 10 and 11, the coefficient of expanded by

is equal to the right hand side of (42). Similarly, we have

which is equivalent to the right hand side of (43) by (25) and simplification. The proof of Lemma 12 is complete.

Proof of Theorems 2 and 3. By using formula (42), the left hand side of (7) is equal to

Similarly, with the help of formula (43), the left hand side of (8) equals

Using formulas (41) and (43) and noticing that the orthogonality of previous two types of -Laguerre polynomials for the case of , we can deduce (7) and (8). The proof of Theorems 2 and 3 is complete.

3. Representations of -Hermite Polynomials

Doha [34, page 5460] deduced the following result by third-order recurrence relation of the coefficients.

Proposition 14 (see [34, equation ]). For and , one has

In this section, we employ the technique of rearrangement of series

to derive the following -analogue of Proposition 14.

Theorem 15. For and , one has

where

and denotes the greatest integer not exceeding .

Theorem 16. For and , one has

where

Before the proof of Theorem 15 the following lemma is necessary.

Lemma 17. For and , one has

Proof. Letting in (38) and using the following fact [8, page 23]:

similarly, we deduce the explicit representation of (39) and (40), respectively,

so we obtain the formula (56). The proof is complete.

Lemma 18 (see [7, equations () and ()]). One has

Proof. By using [7, equations () and ()]

and replacing, respectively, by

we deduce the proof of Lemma 18. The proof is complete.

Proof of Theorem 15. From the generating function of , we have

Comparing the coefficients of on both sides of (62), we obtain the results.

Proof of Theorem 16. From the generating function of , we have

Equating the coefficients of on both sides of (63), we obtain the results.

Proof of Corollary 23. In view of the fact that

we have

Combing (65) and (2), we deduce (50). The proof of Corollary 23 is complete.

4. Integrals Related to Generalized -Hermite Polynomials

The authors [23] deduced the following interesting result inspired by the relation of (16) and the orthogonality of -Laguerre polynomials (6).

Proposition 19 (see [23, Theorem 1]). The sequence of the -polynomials , which are defined by the relations (16), satisfies the orthogonality relation

on the whole real line , where .

In this section, we will further consider multivariate -Hermite polynomials by Theorems 2 and 3.

Theorem 20. For , one has

Theorem 21. For , one has

Remark 22. For and , Theorems 20 and 21 reduce to Proposition 19 and Corollary 23 respectively.

Corollary 23. The sequence of the -polynomials , which are defined by the relations (16), satisfies the orthogonality relation

on the whole real line , where .

Proof of Theorems 20 and 21. Let and represent the right hand side of (7) and (8) respectively. Since the weight function in (67) is an even function of the independent variable by the definition (16), so the polynomials are evidently orthogonal to each other when the degrees and are either simultaneously even or odd. From (16) and Theorem 2 we have

Putting (70) and (71) together and using Theorem 2, we complete the proof of Theorem 20 after some simplification. In the same way we find that

which are two cases of Theorem 21; thus we obtain the proof.

Proof of Proposition 19 and Corollary 23. Let us consider first that the case of both and is even, and just take and in Theorem 20. We have

Putting (73) and (74) together results in the orthogonality relation (66). The proof of Proposition 19 is complete. By the same way, we can deduce Corollary 23. This completes the proof.

Acknowledgments

The author thanks Professor Roelof Koekoek, who provided the hardcopy of his paper kindly. This work was supported by Tianyuan Special Funds of the National Natural Science Foundation of China (no. 11226298), China Postdoctoral Science Foundation (no. 2012M521155), Zhejiang Projects for Postdoctoral Research Preferred Funds (no. Bsh1201021), and Zhejiang Provincial Natural Science Foundation of China (no. LQ13A010021).

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